The paper deals with the heavy traffic modelling and optimal control of trunk line networks. The controls concern either acceptance or rerouting. Both discounted and ergodic cost functions are used. It is shown that the sequence of optimal value functions of the physical process converges to the optimal value function for the limit model. Also, good controls for the limit system are good for the physical system. Single trunk systems with both priority and other inputs as well as networks are addressed. For the network case, the limit model is nonstandard, since the control occurs only on the boundary, and is a control over the direction of reflection.